Do you mean 'to sum up' and 'in conclusion' are used in situations such as a piece of writing or a speech, etc., and they are interchangeable To my mind, yes, they are. Although perhaps "to sum up" is a little less formal.
and 'in a word' and 'in a nutshell' are used in situations such as a short and clear answer, explanation or statement in conversation, etc.? Yes. "What did you think of the movie?" "In a word: dreadful!" "In a nutshell, it was three hours of my life I will never get back." Realize that "in a nutshell" is rather slang-like and informal.
In what situations could we use 'in sum'? Perhaps in either of the two situations above. It's the least useful one of these, I think. Or maybe: "What did you think of the movie?" "It was poorly written, poorly acted, boring, and gross at the same time. In sum, I hated it."